# example of four exponentials conjecture

Taking ${x}_{1}=i\pi $, ${x}_{2}=i\pi \sqrt{2}$, ${y}_{1}=1$, ${y}_{2}=\sqrt{2}$, we see that this conjecture implies that one of ${e}^{i\pi}$, ${e}^{i\pi \sqrt{2}}$, or ${e}^{i2\pi}$ is transcendental. Since the first is $-1$ and the last is $1$, the conjecture states that second must be transcendental, that is, ${e}^{i\pi \sqrt{2}}$ is (conjecturally) transcendental.

In this particular case, the result is known already, so the conjecture is verified. Using Gelfond’s theorem, take $\alpha ={e}^{i\pi}$ and $\beta =\sqrt{2}$ and it follows that ${\alpha}^{\beta}$ is transcendental.

Title | example of four exponentials conjecture^{} |
---|---|

Canonical name | ExampleOfFourExponentialsConjecture |

Date of creation | 2013-03-22 14:09:09 |

Last modified on | 2013-03-22 14:09:09 |

Owner | archibal (4430) |

Last modified by | archibal (4430) |

Numerical id | 6 |

Author | archibal (4430) |

Entry type | Example |

Classification | msc 11J81 |